Although not impossible, finding the operator \(\mathbf {K}\)  “by hand”
(i.e. finding a basis for the left null-space of \(\mathbf {A}\) ) for a
redundant aperture is a tedious task, as the matrix \(\mathbf {A}\)  can
get quite large.
A very efficient way to do this is to calcutate the singular value
decomposition (SVD) of \(\mathbf {A}^T\) .
The SVD algorithm [1]}, allows to decompose the
now \((N-1) \times M\)  matrix \(\mathbf {A}^T\)  as the product of a \((N-1)\times M\)  column-orthogonal matrix \(\mathbf {U}\) , a \(M \times M\) 
diagonal matrix \(\mathbf {W}\)  with positive or zero elements (the
so-called singular values) and the transpose of an \(M \times M\) 
orthogonal matrix \(\mathbf {V}\) :
\(\mathbf {A}^T = \mathbf {U} \cdot \mathbf {W} \cdot \mathbf {V}^T.\) 
